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Cayley diagram

C4×#41⋊C3 #5438

order192
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = A4 · Out(G) = C2×#120
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×#41⋊C3
unambiguous semidirect product(C4×#41)⋊C3

Splittings

Fitting, SylowC4×#41 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC42A4
commutator, hypocenterQ8C2×C3×C4
Frattini, socleC22C22×A4
FittingC4×#41C3
prime radicalC2×#41⋊C3C2
cosocleC2×Q8C22×C3
LCSG ▷ Q8
UCSC1 ◁ C42

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 20 points:

(5,6,9,12)(7,11,15,18,8,17,14,20,13,10,16,19);(1,2,3,4)(5,13,9,15)(6,17,12,19)(7,16,14,8)(10,20,18,11)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72268256, #72268257, #72268987, …
maximal subgroup ofC4×GU(2,𝔽3), #73067, #73068, …
maximal normal subgroup ofC4×GU(2,𝔽3), #73067, #73068, …
maximal quotient ofC42×SL(2,𝔽3), #2341⋊C3, #2351⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 997);